Question

Evaluate:-
The polynomials P(x) = kx³ + 3x² - 3 and Q(x) = 2x³ - 5x +k, when divided by (x-4) leave the same remainder. The value of k is -
(a) 2
(b) 1
(c) 0
(d) -1

Answer 1

We can solve the problem using the method of long division to find the two remainders.

{ Refer to the two attachments to check the remainders. }

• On division of P(x) by (x - 4), we get the remainder as (64k + 45)

• On division of Q(x) by (x - 4), we get the remainder as (k + 108)

By the given conditions,

    64k + 45 = k + 108

or, 64k - k = 108 - 45

or, 63k = 63

or, k = 1

Therefore, option (b) : k = 1 is correct.

Answer 2

Solution:-

Let p(x) = kx³ + 3x² - 3

and q(x) = 2x³ -5x + k

Also, p(4) = 64k + 48 - 3 = 64k + 45

and q(4) = 128 - 20 + k = 108 + k

Thus, p(x) leaves the remainder 64k + 45 and q(x) leaves the remainder 108 + k when divided by 4.

According to question,

p(4) = q(4)

⇒ 64k + 45 = 108 + k

⇒ 63k = 108 - 45

⇒ 63k = 63

⇒ k = 1